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Lines on K3 quartic surfaces in characteristic 3

Davide Cesare Veniani

Abstract

We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the Fermat quartic surface; otherwise, \(X\) contains at most 67 lines. We improve this bound to 58 if \(X\) contains a star (ie four distinct lines intersecting at a smooth point of \(X\)). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

Lines on K3 quartic surfaces in characteristic 3

Abstract

We investigate the number of straight lines contained in a K3 quartic surface defined over an algebraically closed field of characteristic 3. We prove that if contains 112 lines, then is projectively equivalent to the Fermat quartic surface; otherwise, contains at most 67 lines. We improve this bound to 58 if contains a star (ie four distinct lines intersecting at a smooth point of ). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

Paper Structure

This paper contains 21 sections, 38 theorems, 66 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Principal results
  4. Contents of the paper
  5. Preliminaries
  6. Lines and singular points
  7. Valency
  8. Lines of the first and second kind
  9. Triangle-free surfaces
  10. Elliptic fibrations
  11. Quasi-elliptic fibrations and cuspidal lines
  12. Bounds on the valency
  13. Separable elliptic lines
  14. Quasi-elliptic lines of degree 3
  15. Quasi-elliptic lines of lower degree
  16. ...and 6 more sections

Key Result

theorem 1

Assume $\Char \IK = 3$. If a K3 quartic surface $X$ is not projectively equivalent to the Fermat surface, then $|{\Fn X}| \leq 67$.

Figures (5)

  • Figure 1: Possible configurations of lines on a plane with a triangle. Singular points are marked white. In configurations $\mathcal{D}_0$ and $\mathcal{E}_0$ the singular points might coincide.
  • Figure 2: Cuspidal curve (white vertex) intersecting a reducible fiber of a quasi-elliptic fibration. Multiple white dots represent different possibilities.
  • Figure 3: Possible residual cubics corresponding to a fiber of type $\IV$.
  • Figure 4: Possible residual cubics corresponding to a fiber of type $\IV^*$.
  • Figure 5: Possible residual cubics appearing in the proof of \ref{['prop:qe-deg2']}.

Theorems & Definitions (41)

  • theorem 1: see \ref{['subsec:proof.thm:char3']}
  • lemma 1: veniani1
  • lemma 2: veniani1
  • definition 1
  • lemma 3
  • proposition 1: veniani-char2
  • lemma 4: veniani-char2
  • lemma 5: veniani-char2
  • lemma 6
  • lemma 7
  • ...and 31 more